Problem: What is the period of the function $f(x)=-6\sin(3\pi x+4)-2$ ? Give an exact value. units
Solution: Period in sinusoids of the form $y=a\sin(bx+c)+d$ Graphically, the period of a sinusoidal function is the horizontal distance between the ends of a single cycle of its graph. The period of a sinusoid of the form $y={a}\sin( {b}x + c) + {d}$ is equal to $\dfrac{2\pi}{| b|}$. [How can we justify this given our graphical understanding of period?] Finding the period The period of $f(x) = -6\sin({3\pi}x+4)-2$ is: $\begin{aligned} \text{period}&=\dfrac{2\pi}{|{b}|}\\\\ &=\dfrac{2\pi}{| {3\pi}|} \\\\\\\\\\ &= \dfrac{2}{3} \\ \end{aligned}$ The answer The period of $f(x) = -6\sin({3\pi}x+4)-2$ is $\dfrac23$ units.